The last person on Earth most people would want to accidentally bump into on holiday would be their manager from work. But are the chances of that happening greater than you think?

It may seem like a massive coincidence to meet someone you know far from home, and many people have anecdotes of when it has happened to them.

Take the case of Katherine, a young British woman planning a summer holiday abroad with her friends.

"The problem was I couldn't get the time off work. But I decided to phone in sick for the week and go on holiday anyway," she says.

But when she arrived at the resort with her friends, Katherine went out to see the view from their apartment, and there on the balcony above was her boss, from the supermarket where she worked in Weston Super Mare.

"Obviously, she was surprised to see me, as I was supposed to be sick."

What was the likelihood of such a coincidence?

Katie Chicot, a mathematician from the UK's Open University, has looked at the probabilities involved.

The lesson here is, if you don't want to go on the same holiday as your boss, don't use the same travel agentKatie Chicot, Open University

She started by asking the question - what are the chances that Katherine books a holiday in the same place and at the same time as her boss?

"Let's assume that they are independent events, they both want to choose a nice sunny summer holiday and they both want a cheap holiday.

"They don't have children, so they are going to avoid the school holidays," Chicot says.

She makes the assumption that there are just seven weeks of summer to choose from when you exclude the school holidays running from the end of July to the end of August.

As well as working out the probability of picking the same week, she then looks at the chances of Katherine picking the same destination.

She says Spain is the most popular holiday destination for Brits.

Image caption
Sun, sea, line manager...

"There is about a one-in-three chance of choosing Spain, so the chances of choosing the same week
and choosing Spain, are one in 20," she calculates.

But it turns out there is a travel agent near the supermarket where Katherine and her boss worked.

"This UK travel agency sends 800,000 people a year to Spain, of which 20,000 go to Magaluf - the resort where Katherine was caught red-handed.

"Using these figures, the relative probability that a person selects Magaluf, is about two in 100."

Surprise!

Statistician Byron Jones explains why if you meet 22 strangers at a party there is a 50-50 chance that two of you share the same birthday.

First we'll calculate the probability that you all 23 of you have a different birthday. If you're no.1, the chance that no.2 has a different birthday is 364/365. The chance that no.3 has a different birthday from 1 and 2 is 363/365. So the chance that 1 to 3 have different birthdays is (364/365) x (363/365).

The chance that all 23 have different birthdays is (364/365) x (363/365) x... x (343/365) = 0.493

The chance that at least one pair of strangers shares the same birthday is 1 minus this number, or 0.507... just greater than 50-50.

To recap, there is a one in three chance of Katherine choosing Spain, and if she does, there is a two in 100 chance of choosing Magaluf.

"So the chance of Katherine selecting Magaluf, all things being equal, are about eight in a 1,000," says Chicot.

But as noted above there is a one-in-seven chance of Katherine and her boss selecting the same week, so "the chance that she selects the same week, and Magaluf, is about one in a 1,000," says Chicot.

That's if we assume that the events of Katherine choosing Magaluf and her boss choosing Magaluf are independent events.

But, of course, Chicot suspects that they are not independent, as they probably used the same travel agent.

"At any one time, a travel agent will have the same best deal on offer. So if two people walk into the same travel agent within a similar period of time and ask for similar holiday requirements, it's very likely they are going to be offered the same holiday."

So the chances of this happening to Katherine are not as remote as one in 1,000.

"And the lesson here is, if you don't want to go on the same holiday as your boss, don't use the same travel agent as them," says Chicot.

Coincidences capture people's imagination. They might believe it is fate, but statisticians, such as Byron Jones, based in Switzerland, would call them "the law of truly large numbers".

"We tend to personalise coincidences, and when we meet a friend in some distant location unexpectedly, we tend to think that's very surprising," he says.

More or Less: Behind the stats

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"People look into it more deeply to try to see an explanation for it, and what we don't really appreciate is that friends are bumping into each other globally all the time, and that event in itself is not that unusual."

In a sense, it's like the lottery - it would be very surprising if it happened to you, but it's not at all surprising that it happens to someone.

But not only are "coincidences" happening all the time, Byron Jones says that humans often misunderstand statistics and probability, assuming events are rarer than they are.

In 1986, The New York Times reported that a woman from New Jersey had won the state lottery twice. It was reported that this was a one in 17 trillion chance - a statistic that came in for criticism at the time.

Byron Jones has done some calculations with a colleague, Robb Muirhead, and they calculated that while the chance of a specific individual winning twice was exceedingly low, the chances of
someone winning twice, over a seven-year period, was about 50-50, making it a much more likely event than people might assume.